#7633313

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2,3,4)(6,10)(11,12)(13,14)(15,24)(16,20)(17,21)(18,22)(19,23)(25,28)(26,27);(1,2)(3,4)(13,15,19,21)(14,17,23,24)(16,26,18,25)(20,28,22,27);(5,6,9,10)(7,11,8,12)(13,16,19,18)(14,20,23,22)(15,25,21,26)(17,27,24,28);(5,7,9,8)(6,11,10,12)(13,14)(15,17)(16,20)(18,22)(19,23)(21,24)(25,27)(26,28);(5,7,9,8)(6,11,10,12)(13,17)(14,21)(15,23)(16,27)(18,28)(19,24)(20,26)(22,25);(5,8,9,7)(6,12,10,11)(13,18,19,16)(14,22,23,20)(15,26,21,25)(17,28,24,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540509);